R Notebook
library(tidyverse)
library(car)
library(modelr)
library(skimr)
library(GGally)
library(ggfortify)Question 1
Tidy up the data ready for regression:
house <- read_csv("data/kc_house_data.csv") %>% janitor::clean_names()Rows: 21613 Columns: 21
── Column specification ──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Delimiter: ","
chr (1): id
dbl (19): price, bedrooms, bathrooms, sqft_living, sqft_lot, floors, waterfront, view, condition, grade, sqft_above, sqft_basement, y...
dttm (1): date
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.summary(house) id date price bedrooms bathrooms sqft_living sqft_lot
Length:21613 Min. :2014-05-02 00:00:00.00 Min. : 75000 Min. : 0.000 Min. :0.000 Min. : 290 Min. : 520
Class :character 1st Qu.:2014-07-22 00:00:00.00 1st Qu.: 321950 1st Qu.: 3.000 1st Qu.:1.750 1st Qu.: 1427 1st Qu.: 5040
Mode :character Median :2014-10-16 00:00:00.00 Median : 450000 Median : 3.000 Median :2.250 Median : 1910 Median : 7618
Mean :2014-10-29 04:38:01.96 Mean : 540088 Mean : 3.371 Mean :2.115 Mean : 2080 Mean : 15107
3rd Qu.:2015-02-17 00:00:00.00 3rd Qu.: 645000 3rd Qu.: 4.000 3rd Qu.:2.500 3rd Qu.: 2550 3rd Qu.: 10688
Max. :2015-05-27 00:00:00.00 Max. :7700000 Max. :33.000 Max. :8.000 Max. :13540 Max. :1651359
floors waterfront view condition grade sqft_above sqft_basement yr_built
Min. :1.000 Min. :0.000000 Min. :0.0000 Min. :1.000 Min. : 1.000 Min. : 290 Min. : 0.0 Min. :1900
1st Qu.:1.000 1st Qu.:0.000000 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.: 7.000 1st Qu.:1190 1st Qu.: 0.0 1st Qu.:1951
Median :1.500 Median :0.000000 Median :0.0000 Median :3.000 Median : 7.000 Median :1560 Median : 0.0 Median :1975
Mean :1.494 Mean :0.007542 Mean :0.2343 Mean :3.409 Mean : 7.657 Mean :1788 Mean : 291.5 Mean :1971
3rd Qu.:2.000 3rd Qu.:0.000000 3rd Qu.:0.0000 3rd Qu.:4.000 3rd Qu.: 8.000 3rd Qu.:2210 3rd Qu.: 560.0 3rd Qu.:1997
Max. :3.500 Max. :1.000000 Max. :4.0000 Max. :5.000 Max. :13.000 Max. :9410 Max. :4820.0 Max. :2015
yr_renovated zipcode lat long sqft_living15 sqft_lot15
Min. : 0.0 Min. :98001 Min. :47.16 Min. :-122.5 Min. : 399 Min. : 651
1st Qu.: 0.0 1st Qu.:98033 1st Qu.:47.47 1st Qu.:-122.3 1st Qu.:1490 1st Qu.: 5100
Median : 0.0 Median :98065 Median :47.57 Median :-122.2 Median :1840 Median : 7620
Mean : 84.4 Mean :98078 Mean :47.56 Mean :-122.2 Mean :1987 Mean : 12768
3rd Qu.: 0.0 3rd Qu.:98118 3rd Qu.:47.68 3rd Qu.:-122.1 3rd Qu.:2360 3rd Qu.: 10083
Max. :2015.0 Max. :98199 Max. :47.78 Max. :-121.3 Max. :6210 Max. :871200 skim(house)── Data Summary ────────────────────────
Values
Name house
Number of rows 21613
Number of columns 21
_______________________
Column type frequency:
character 1
numeric 19
POSIXct 1
________________________
Group variables None remove unwanted columns
house <- house %>%
select(-c(date, id, sqft_living15, sqft_lot15, zipcode))change waterfront and yr_renovated into logical columns
house <- house %>%
mutate(waterfront = as.logical(waterfront),
yr_renovated = if_else(yr_renovated > 0, TRUE, FALSE))convert view, condition and grade to ordinal type
view_level <- house %>%
distinct(view) %>%
arrange(view) %>%
pull()
condition_level <- house %>%
distinct(condition) %>%
arrange(condition) %>%
pull()
grade_level <- house %>%
distinct(grade) %>%
arrange(grade) %>%
pull()
house <- house %>%
mutate(view = factor(view, levels = view_level, ordered = TRUE),
condition = factor(condition, levels = condition_level, ordered = TRUE),
grade = factor(grade, levels = grade_level, ordered = TRUE))skim(house)── Data Summary ────────────────────────
Values
Name house
Number of rows 21613
Number of columns 16
_______________________
Column type frequency:
factor 3
logical 2
numeric 11
________________________
Group variables None house %>%
count(view)
house %>%
count(condition)
house %>%
count(grade)Question 2
Check for aliased variables using the alias() function (this takes in a formula object and a data set).
check for alias using alias
alias(lm(price ~ .,
data = house))Model :
price ~ bedrooms + bathrooms + sqft_living + sqft_lot + floors +
waterfront + view + condition + grade + sqft_above + sqft_basement +
yr_built + yr_renovated + lat + long
Complete :
(Intercept) bedrooms bathrooms sqft_living sqft_lot floors waterfrontTRUE view.L view.Q view.C view^4 condition.L
sqft_basement 0 0 0 1 0 0 0 0 0 0 0 0
condition.Q condition.C condition^4 grade.L grade.Q grade.C grade^4 grade^5 grade^6 grade^7 grade^8 grade^9 grade^10
sqft_basement 0 0 0 0 0 0 0 0 0 0 0 0 0
grade^11 sqft_above yr_built yr_renovatedTRUE lat long
sqft_basement 0 -1 0 0 0 0 remove aliased variable from the data and check again for aliases
house <- house %>%
select(- sqft_basement)
alias(lm(price ~ .,
data = house))Model :
price ~ bedrooms + bathrooms + sqft_living + sqft_lot + floors +
waterfront + view + condition + grade + sqft_above + yr_built +
yr_renovated + lat + longQuestion 3
Systematically build a regression model containing up to four main effects (remember, a main effect is just a single predictor with coefficient), testing the regression diagnostics as you go * splitting datasets into numeric and non-numeric columns might help ggpairs() run in manageable time, although you will need to add either a price or resid column to the non-numeric dataframe in order to see its correlations with the non-numeric predictors.
select all numeric types
house_numeric <- house %>%
select_if(is.numeric)select all non-numeric types
house_nonnumeric <- house %>%
select_if(function(x) !is.numeric(x))add in price column from house data
house_nonnumeric$price <- house$priceadd ln_price in all data sets
house <- house %>%
mutate(ln_price = log(price))
house_numeric <- house_numeric %>%
mutate(ln_price = log(price))
house_nonnumeric <- house_nonnumeric %>%
mutate(ln_price = log(price))ggpair the numeric data
house_numeric %>%
select(price, everything()) %>%
ggpairs()
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house_numeric %>%
select(ln_price, everything()) %>%
ggpairs()
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investigate:
- sqft living
- sqft above
- bathrooms
- bedrooms
house_nonnumeric %>%
select(price, everything()) %>%
ggpairs()
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house_nonnumeric %>%
select(ln_price, everything()) %>%
ggpairs()
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investigate:
- grade
- waterfront
- view
- condition
First model
mod1a1 <- lm(ln_price ~ sqft_living,
data = house)
summary(mod1a1)
Call:
lm(formula = ln_price ~ sqft_living, data = house)
Residuals:
Min 1Q Median 3Q Max
-2.97781 -0.28543 0.01472 0.26070 1.27628
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.222e+01 6.374e-03 1916.9 <2e-16 ***
sqft_living 3.987e-04 2.803e-06 142.2 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3785 on 21611 degrees of freedom
Multiple R-squared: 0.4835, Adjusted R-squared: 0.4835
F-statistic: 2.023e+04 on 1 and 21611 DF, p-value: < 2.2e-16autoplot(mod1a1)
mod1a2 <- lm(price ~ sqft_living,
data = house)
summary(mod1a2)
Call:
lm(formula = price ~ sqft_living, data = house)
Residuals:
Min 1Q Median 3Q Max
-1476062 -147486 -24043 106182 4362067
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -43580.743 4402.690 -9.899 <2e-16 ***
sqft_living 280.624 1.936 144.920 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 261500 on 21611 degrees of freedom
Multiple R-squared: 0.4929, Adjusted R-squared: 0.4928
F-statistic: 2.1e+04 on 1 and 21611 DF, p-value: < 2.2e-16autoplot(mod1a2)
mod1b1 <- lm(ln_price ~ grade,
data = house)
summary(mod1b1)
Call:
lm(formula = ln_price ~ grade, data = house)
Residuals:
Min 1Q Median 3Q Max
-1.42892 -0.26231 -0.00483 0.24505 1.80117
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 13.159997 0.037667 349.378 < 2e-16 ***
grade.L 3.380842 0.197635 17.106 < 2e-16 ***
grade.Q 0.565833 0.200908 2.816 0.00486 **
grade.C -0.012459 0.180001 -0.069 0.94482
grade^4 0.014025 0.159772 0.088 0.93005
grade^5 0.122204 0.142440 0.858 0.39094
grade^6 -0.019036 0.120748 -0.158 0.87473
grade^7 0.010765 0.093391 0.115 0.90823
grade^8 -0.007330 0.064806 -0.113 0.90995
grade^9 -0.004064 0.039929 -0.102 0.91893
grade^10 0.004848 0.021378 0.227 0.82060
grade^11 0.015221 0.009383 1.622 0.10476
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3735 on 21601 degrees of freedom
Multiple R-squared: 0.4973, Adjusted R-squared: 0.4971
F-statistic: 1943 on 11 and 21601 DF, p-value: < 2.2e-16autoplot(mod1b1)Warning: Removed 275 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
mod1b2 <- lm(price ~ grade,
data = house)
summary(mod1b2)
Call:
lm(formula = price ~ grade, data = house)
Residuals:
Min 1Q Median 3Q Max
-1929615 -135853 -35090 89080 5565658
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 941742 25666 36.693 < 2e-16 ***
grade.L 3000531 134665 22.281 < 2e-16 ***
grade.Q 1703928 136895 12.447 < 2e-16 ***
grade.C 757073 122650 6.173 6.84e-10 ***
grade^4 314354 108866 2.888 0.00389 **
grade^5 198604 97056 2.046 0.04074 *
grade^6 78933 82275 0.959 0.33738
grade^7 39242 63635 0.617 0.53746
grade^8 2243 44158 0.051 0.95948
grade^9 2531 27207 0.093 0.92589
grade^10 1258 14567 0.086 0.93116
grade^11 8633 6393 1.350 0.17694
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 254500 on 21601 degrees of freedom
Multiple R-squared: 0.5197, Adjusted R-squared: 0.5195
F-statistic: 2125 on 11 and 21601 DF, p-value: < 2.2e-16autoplot(mod1b2)Warning: Removed 2313 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
Choose sqft_living as first predictor
Look at residuals
house_resid_numeric_log <- house_numeric %>%
add_residuals(mod1a1) %>%
select(-c(ln_price, sqft_living))
house_resid_numeric_log %>%
select(resid, everything()) %>%
ggpairs()
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- lat
house_resid_numeric <- house_numeric %>%
add_residuals(mod1a2) %>%
select(-c(price, sqft_living))
house_resid_numeric %>%
select(resid, everything()) %>%
ggpairs()
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- bedrooms
house_resid <- house %>%
add_residuals(mod1a1) %>%
select(-c(ln_price, sqft_living))
house_resid %>%
select(resid, everything()) %>%
ggpairs()
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- waterfront
- grade
- condition
Second predictor
mod2a1 <- lm(ln_price ~ sqft_living + lat,
data = house)
summary(mod2a1)
Call:
lm(formula = ln_price ~ sqft_living + lat, data = house)
Residuals:
Min 1Q Median 3Q Max
-3.00396 -0.19459 -0.01122 0.18293 1.24316
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -6.256e+01 7.239e-01 -86.42 <2e-16 ***
sqft_living 3.863e-04 2.297e-06 168.17 <2e-16 ***
lat 1.573e+00 1.522e-02 103.31 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3097 on 21610 degrees of freedom
Multiple R-squared: 0.6542, Adjusted R-squared: 0.6542
F-statistic: 2.045e+04 on 2 and 21610 DF, p-value: < 2.2e-16autoplot(mod2a1)
mod2a2 <- lm(price ~ sqft_living + lat,
data = house)
summary(mod2a2)
Call:
lm(formula = price ~ sqft_living + lat, data = house)
Residuals:
Min 1Q Median 3Q Max
-1487994 -125643 -20309 84613 4368717
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.416e+07 5.653e+05 -60.44 <2e-16 ***
sqft_living 2.749e+02 1.794e+00 153.27 <2e-16 ***
lat 7.177e+05 1.189e+04 60.36 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 241900 on 21610 degrees of freedom
Multiple R-squared: 0.566, Adjusted R-squared: 0.566
F-statistic: 1.409e+04 on 2 and 21610 DF, p-value: < 2.2e-16autoplot(mod2a2)
mod2b1 <- lm(ln_price ~ sqft_living + waterfront,
data = house)
summary(mod2b1)
Call:
lm(formula = ln_price ~ sqft_living + waterfront, data = house)
Residuals:
Min 1Q Median 3Q Max
-2.90243 -0.28303 0.01682 0.26106 1.27143
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.223e+01 6.320e-03 1934.61 <2e-16 ***
sqft_living 3.926e-04 2.790e-06 140.73 <2e-16 ***
waterfrontTRUE 6.301e-01 2.961e-02 21.28 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3746 on 21610 degrees of freedom
Multiple R-squared: 0.4941, Adjusted R-squared: 0.4941
F-statistic: 1.055e+04 on 2 and 21610 DF, p-value: < 2.2e-16autoplot(mod2b1)
mod2b2 <- lm(price ~ sqft_living + waterfront,
data = house)
summary(mod2b2)
Call:
lm(formula = price ~ sqft_living + waterfront, data = house)
Residuals:
Min 1Q Median 3Q Max
-1376782 -142867 -21360 107201 4449253
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -32957.851 4242.971 -7.768 8.35e-15 ***
sqft_living 272.507 1.873 145.499 < 2e-16 ***
waterfrontTRUE 829983.104 19882.279 41.745 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 251500 on 21610 degrees of freedom
Multiple R-squared: 0.5307, Adjusted R-squared: 0.5307
F-statistic: 1.222e+04 on 2 and 21610 DF, p-value: < 2.2e-16autoplot(mod2b2)
mod2c1 <- lm(ln_price ~ sqft_living + grade,
data = house)
summary(mod2c1)
Call:
lm(formula = ln_price ~ sqft_living + grade, data = house)
Residuals:
Min 1Q Median 3Q Max
-1.63935 -0.25462 -0.00143 0.23511 1.41830
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.259e+01 3.700e-02 340.222 <2e-16 ***
sqft_living 2.194e-04 4.110e-06 53.370 <2e-16 ***
grade.L 1.846e+00 1.880e-01 9.819 <2e-16 ***
grade.Q 5.663e-02 1.891e-01 0.300 0.765
grade.C -1.518e-01 1.692e-01 -0.897 0.370
grade^4 -4.596e-02 1.502e-01 -0.306 0.760
grade^5 5.557e-02 1.339e-01 0.415 0.678
grade^6 -3.328e-02 1.135e-01 -0.293 0.769
grade^7 -2.024e-03 8.778e-02 -0.023 0.982
grade^8 -3.063e-03 6.092e-02 -0.050 0.960
grade^9 -2.060e-02 3.753e-02 -0.549 0.583
grade^10 1.083e-02 2.010e-02 0.539 0.590
grade^11 -1.707e-03 8.825e-03 -0.193 0.847
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3511 on 21600 degrees of freedom
Multiple R-squared: 0.5559, Adjusted R-squared: 0.5557
F-statistic: 2253 on 12 and 21600 DF, p-value: < 2.2e-16autoplot(mod2c1)Warning: Removed 1228 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
anova(mod1a1, mod2c1)Analysis of Variance Table
Model 1: ln_price ~ sqft_living
Model 2: ln_price ~ sqft_living + grade
Res.Df RSS Df Sum of Sq F Pr(>F)
1 21611 3096.5
2 21600 2662.4 11 434.12 320.18 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1mod2c2 <- lm(price ~ sqft_living + grade,
data = house)
summary(mod2c2)
Call:
lm(formula = price ~ sqft_living + grade, data = house)
Residuals:
Min 1Q Median 3Q Max
-1550528 -129352 -27914 92324 4677732
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.310e+05 2.503e+04 21.214 < 2e-16 ***
sqft_living 1.573e+02 2.781e+00 56.567 < 2e-16 ***
grade.L 1.900e+06 1.272e+05 14.938 < 2e-16 ***
grade.Q 1.339e+06 1.279e+05 10.466 < 2e-16 ***
grade.C 6.572e+05 1.145e+05 5.740 9.57e-09 ***
grade^4 2.713e+05 1.016e+05 2.671 0.00758 **
grade^5 1.508e+05 9.058e+04 1.665 0.09593 .
grade^6 6.872e+04 7.679e+04 0.895 0.37083
grade^7 3.007e+04 5.939e+04 0.506 0.61262
grade^8 5.303e+03 4.121e+04 0.129 0.89761
grade^9 -9.327e+03 2.539e+04 -0.367 0.71339
grade^10 5.547e+03 1.360e+04 0.408 0.68327
grade^11 -3.506e+03 5.971e+03 -0.587 0.55708
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 237500 on 21600 degrees of freedom
Multiple R-squared: 0.5817, Adjusted R-squared: 0.5814
F-statistic: 2503 on 12 and 21600 DF, p-value: < 2.2e-16autoplot(mod2c2)Warning: Removed 4665 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
anova(mod1a2, mod2c2)Analysis of Variance Table
Model 1: price ~ sqft_living
Model 2: price ~ sqft_living + grade
Res.Df RSS Df Sum of Sq F Pr(>F)
1 21611 1.4773e+15
2 21600 1.2186e+15 11 2.5871e+14 416.89 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1mod2d1 <- lm(ln_price ~ sqft_living + condition,
data = house)
summary(mod2d1)
Call:
lm(formula = ln_price ~ sqft_living + condition, data = house)
Residuals:
Min 1Q Median 3Q Max
-2.96468 -0.28308 0.01632 0.25785 1.45271
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.215e+01 1.580e-02 769.081 < 2e-16 ***
sqft_living 3.993e-04 2.802e-06 142.469 < 2e-16 ***
condition.L 2.997e-01 4.475e-02 6.697 2.19e-11 ***
condition.Q 2.876e-02 3.783e-02 0.760 0.4472
condition.C -5.404e-02 2.858e-02 -1.891 0.0586 .
condition^4 8.532e-02 1.636e-02 5.217 1.84e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3756 on 21607 degrees of freedom
Multiple R-squared: 0.4915, Adjusted R-squared: 0.4914
F-statistic: 4177 on 5 and 21607 DF, p-value: < 2.2e-16autoplot(mod2d1)
mod2d2 <- lm(price ~ sqft_living + grade,
data = house)
summary(mod2d2)
Call:
lm(formula = price ~ sqft_living + grade, data = house)
Residuals:
Min 1Q Median 3Q Max
-1550528 -129352 -27914 92324 4677732
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.310e+05 2.503e+04 21.214 < 2e-16 ***
sqft_living 1.573e+02 2.781e+00 56.567 < 2e-16 ***
grade.L 1.900e+06 1.272e+05 14.938 < 2e-16 ***
grade.Q 1.339e+06 1.279e+05 10.466 < 2e-16 ***
grade.C 6.572e+05 1.145e+05 5.740 9.57e-09 ***
grade^4 2.713e+05 1.016e+05 2.671 0.00758 **
grade^5 1.508e+05 9.058e+04 1.665 0.09593 .
grade^6 6.872e+04 7.679e+04 0.895 0.37083
grade^7 3.007e+04 5.939e+04 0.506 0.61262
grade^8 5.303e+03 4.121e+04 0.129 0.89761
grade^9 -9.327e+03 2.539e+04 -0.367 0.71339
grade^10 5.547e+03 1.360e+04 0.408 0.68327
grade^11 -3.506e+03 5.971e+03 -0.587 0.55708
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 237500 on 21600 degrees of freedom
Multiple R-squared: 0.5817, Adjusted R-squared: 0.5814
F-statistic: 2503 on 12 and 21600 DF, p-value: < 2.2e-16autoplot(mod2d2)Warning: Removed 4665 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
Choose lat as second predictor
house_resid_numeric <- house_numeric %>%
add_residuals(mod2a1) %>%
select(-c(ln_price, sqft_living, lat))
house_resid_numeric %>%
select(resid, everything()) %>%
ggpairs()
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- yr_built
house_resid <- house %>%
add_residuals(mod2a1) %>%
select(-c(ln_price, sqft_living, lat))
house_resid %>%
select(resid, everything()) %>%
ggpairs()
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- grade
- waterfront
- condition
third predictor
mod3a1 <- lm(ln_price ~ sqft_living + lat + yr_built,
data = house)
autoplot(mod3a1)
summary(mod3a1)
Call:
lm(formula = ln_price ~ sqft_living + lat + yr_built, data = house)
Residuals:
Min 1Q Median 3Q Max
-3.13482 -0.19229 -0.00225 0.18551 1.27495
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.718e+01 7.686e-01 -74.40 <2e-16 ***
sqft_living 4.018e-04 2.412e-06 166.59 <2e-16 ***
lat 1.521e+00 1.533e-02 99.22 <2e-16 ***
yr_built -1.487e-03 7.616e-05 -19.53 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.307 on 21609 degrees of freedom
Multiple R-squared: 0.6602, Adjusted R-squared: 0.6602
F-statistic: 1.4e+04 on 3 and 21609 DF, p-value: < 2.2e-16mod3b1 <- lm(ln_price ~ sqft_living + lat + grade,
data = house)
autoplot(mod3b1)Warning: Removed 2702 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
summary(mod3b1)
Call:
lm(formula = ln_price ~ sqft_living + lat + grade, data = house)
Residuals:
Min 1Q Median 3Q Max
-1.88559 -0.18372 -0.01666 0.16834 1.33509
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.768e+01 6.752e-01 -85.417 <2e-16 ***
sqft_living 2.371e-04 3.358e-06 70.594 <2e-16 ***
lat 1.477e+00 1.418e-02 104.163 <2e-16 ***
grade.L 1.423e+00 1.534e-01 9.277 <2e-16 ***
grade.Q 3.446e-02 1.543e-01 0.223 0.823
grade.C -1.060e-01 1.381e-01 -0.768 0.443
grade^4 -1.770e-01 1.225e-01 -1.444 0.149
grade^5 1.556e-01 1.092e-01 1.424 0.154
grade^6 -1.507e-01 9.261e-02 -1.627 0.104
grade^7 6.952e-02 7.162e-02 0.971 0.332
grade^8 -6.991e-02 4.970e-02 -1.407 0.160
grade^9 1.601e-02 3.062e-02 0.523 0.601
grade^10 -5.606e-03 1.640e-02 -0.342 0.732
grade^11 3.512e-03 7.200e-03 0.488 0.626
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2864 on 21599 degrees of freedom
Multiple R-squared: 0.7044, Adjusted R-squared: 0.7042
F-statistic: 3959 on 13 and 21599 DF, p-value: < 2.2e-16anova(mod2a1, mod3b1)Analysis of Variance Table
Model 1: ln_price ~ sqft_living + lat
Model 2: ln_price ~ sqft_living + lat + grade
Res.Df RSS Df Sum of Sq F Pr(>F)
1 21610 2072.8
2 21599 1772.2 11 300.66 333.14 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1mod3c1 <- lm(ln_price ~ sqft_living + lat + waterfront,
data = house)
autoplot(mod3c1)
summary(mod3c1)
Call:
lm(formula = ln_price ~ sqft_living + lat + waterfront, data = house)
Residuals:
Min 1Q Median 3Q Max
-2.92268 -0.19151 -0.00763 0.18383 1.23200
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -6.295e+01 7.110e-01 -88.54 <2e-16 ***
sqft_living 3.796e-04 2.268e-06 167.35 <2e-16 ***
lat 1.581e+00 1.495e-02 105.74 <2e-16 ***
waterfrontTRUE 6.807e-01 2.405e-02 28.31 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3041 on 21609 degrees of freedom
Multiple R-squared: 0.6666, Adjusted R-squared: 0.6666
F-statistic: 1.44e+04 on 3 and 21609 DF, p-value: < 2.2e-16mod3d1 <- lm(ln_price ~ sqft_living + lat + condition,
data = house)
autoplot(mod3d1)
summary(mod3d1)
Call:
lm(formula = ln_price ~ sqft_living + lat + condition, data = house)
Residuals:
Min 1Q Median 3Q Max
-2.99914 -0.19553 -0.00896 0.18317 1.35340
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -6.269e+01 7.173e-01 -87.393 <2e-16 ***
sqft_living 3.880e-04 2.288e-06 169.615 <2e-16 ***
lat 1.574e+00 1.508e-02 104.352 <2e-16 ***
condition.L 3.015e-01 3.649e-02 8.261 <2e-16 ***
condition.Q -1.629e-02 3.085e-02 -0.528 0.5974
condition.C -3.095e-02 2.330e-02 -1.328 0.1842
condition^4 4.212e-02 1.334e-02 3.156 0.0016 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3063 on 21606 degrees of freedom
Multiple R-squared: 0.6619, Adjusted R-squared: 0.6618
F-statistic: 7050 on 6 and 21606 DF, p-value: < 2.2e-16choose grade
look at residuals
house_resid <- house %>%
add_residuals(mod3b1) %>%
select(-c(ln_price, sqft_living, lat, grade))
house_resid %>%
select(resid, everything()) %>%
ggpairs()
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- yr_built
- waterfront
- condition
- view
mod4a1 <- lm(ln_price ~ sqft_living + lat + grade + yr_built,
data = house)
autoplot(mod4a1)Warning: Removed 2919 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
summary(mod4a1)
Call:
lm(formula = ln_price ~ sqft_living + lat + grade + yr_built,
data = house)
Residuals:
Min 1Q Median 3Q Max
-1.82744 -0.17028 -0.00154 0.16590 1.27592
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.269e+01 7.053e-01 -60.527 <2e-16 ***
sqft_living 2.343e-04 3.178e-06 73.718 <2e-16 ***
lat 1.315e+00 1.380e-02 95.327 <2e-16 ***
grade.L 1.688e+00 1.453e-01 11.616 <2e-16 ***
grade.Q 4.864e-02 1.460e-01 0.333 0.7390
grade.C -2.733e-01 1.307e-01 -2.091 0.0365 *
grade^4 -1.403e-01 1.160e-01 -1.210 0.2265
grade^5 1.620e-01 1.034e-01 1.567 0.1172
grade^6 -1.666e-01 8.764e-02 -1.901 0.0573 .
grade^7 3.879e-02 6.779e-02 0.572 0.5672
grade^8 -4.769e-02 4.704e-02 -1.014 0.3107
grade^9 1.368e-02 2.898e-02 0.472 0.6369
grade^10 -6.882e-03 1.552e-02 -0.444 0.6574
grade^11 6.962e-03 6.815e-03 1.022 0.3070
yr_built -3.710e-03 7.392e-05 -50.193 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2711 on 21598 degrees of freedom
Multiple R-squared: 0.7353, Adjusted R-squared: 0.7351
F-statistic: 4285 on 14 and 21598 DF, p-value: < 2.2e-16mod4b1 <- lm(ln_price ~ sqft_living + lat + grade + waterfront,
data = house)
autoplot(mod4b1)Warning: Removed 2332 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
summary(mod4b1)
Call:
lm(formula = ln_price ~ sqft_living + lat + grade + waterfront,
data = house)
Residuals:
Min 1Q Median 3Q Max
-1.76309 -0.18007 -0.01341 0.16778 1.34215
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.807e+01 6.613e-01 -87.821 <2e-16 ***
sqft_living 2.321e-04 3.292e-06 70.500 <2e-16 ***
lat 1.485e+00 1.389e-02 106.962 <2e-16 ***
grade.L 1.414e+00 1.502e-01 9.410 <2e-16 ***
grade.Q 3.597e-02 1.511e-01 0.238 0.8118
grade.C -8.623e-02 1.352e-01 -0.638 0.5235
grade^4 -1.381e-01 1.200e-01 -1.151 0.2499
grade^5 1.983e-01 1.070e-01 1.853 0.0639 .
grade^6 -1.274e-01 9.068e-02 -1.405 0.1600
grade^7 8.760e-02 7.013e-02 1.249 0.2117
grade^8 -5.796e-02 4.867e-02 -1.191 0.2337
grade^9 1.609e-02 2.999e-02 0.537 0.5915
grade^10 -5.348e-04 1.606e-02 -0.033 0.9734
grade^11 4.405e-03 7.050e-03 0.625 0.5321
waterfrontTRUE 6.790e-01 2.226e-02 30.508 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2805 on 21598 degrees of freedom
Multiple R-squared: 0.7166, Adjusted R-squared: 0.7164
F-statistic: 3901 on 14 and 21598 DF, p-value: < 2.2e-16mod4c1 <- lm(ln_price ~ sqft_living + lat + grade + condition,
data = house)
autoplot(mod4c1)Warning: Removed 2437 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
summary(mod4c1)
Call:
lm(formula = ln_price ~ sqft_living + lat + grade + condition,
data = house)
Residuals:
Min 1Q Median 3Q Max
-1.78842 -0.18177 -0.01093 0.16237 1.28900
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.752e+01 6.588e-01 -87.307 < 2e-16 ***
sqft_living 2.277e-04 3.282e-06 69.379 < 2e-16 ***
lat 1.474e+00 1.383e-02 106.585 < 2e-16 ***
grade.L 1.470e+00 1.516e-01 9.697 < 2e-16 ***
grade.Q 1.293e-01 1.525e-01 0.848 0.397
grade.C -1.595e-01 1.364e-01 -1.169 0.242
grade^4 -1.486e-01 1.208e-01 -1.230 0.219
grade^5 1.478e-01 1.074e-01 1.377 0.169
grade^6 -1.278e-01 9.074e-02 -1.409 0.159
grade^7 4.612e-02 7.000e-02 0.659 0.510
grade^8 -4.994e-02 4.849e-02 -1.030 0.303
grade^9 6.145e-03 2.984e-02 0.206 0.837
grade^10 1.857e-03 1.597e-02 0.116 0.907
grade^11 1.109e-03 7.015e-03 0.158 0.874
condition.L 2.292e-01 3.403e-02 6.735 1.69e-11 ***
condition.Q 6.787e-02 2.870e-02 2.365 0.018 *
condition.C -2.920e-02 2.147e-02 -1.360 0.174
condition^4 4.729e-03 1.224e-02 0.386 0.699
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.279 on 21595 degrees of freedom
Multiple R-squared: 0.7196, Adjusted R-squared: 0.7194
F-statistic: 3260 on 17 and 21595 DF, p-value: < 2.2e-16mod4d1 <- lm(ln_price ~ sqft_living + lat + grade + view,
data = house)
autoplot(mod4d1)Warning: Removed 2743 row(s) containing missing values (geom_path).
Warning: Removed 1 rows containing missing values (geom_point).
Warning: Removed 1 row(s) containing missing values (geom_path).
summary(mod4d1)
Call:
lm(formula = ln_price ~ sqft_living + lat + grade + view, data = house)
Residuals:
Min 1Q Median 3Q Max
-2.05330 -0.17372 -0.01064 0.16413 1.16842
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.767e+01 6.485e-01 -88.919 < 2e-16 ***
sqft_living 2.189e-04 3.253e-06 67.294 < 2e-16 ***
lat 1.481e+00 1.362e-02 108.786 < 2e-16 ***
grade.L 1.346e+00 1.473e-01 9.138 < 2e-16 ***
grade.Q -2.585e-02 1.481e-01 -0.175 0.86147
grade.C -1.219e-01 1.325e-01 -0.920 0.35754
grade^4 -1.478e-01 1.176e-01 -1.256 0.20902
grade^5 1.796e-01 1.049e-01 1.712 0.08691 .
grade^6 -1.440e-01 8.891e-02 -1.620 0.10525
grade^7 9.201e-02 6.876e-02 1.338 0.18088
grade^8 -6.985e-02 4.772e-02 -1.464 0.14327
grade^9 1.997e-02 2.940e-02 0.679 0.49708
grade^10 -7.697e-03 1.574e-02 -0.489 0.62493
grade^11 6.455e-03 6.913e-03 0.934 0.35041
view.L 3.256e-01 1.184e-02 27.491 < 2e-16 ***
view.Q 3.219e-02 1.090e-02 2.952 0.00316 **
view.C 1.357e-01 1.322e-02 10.260 < 2e-16 ***
view^4 -4.492e-02 1.140e-02 -3.940 8.17e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.275 on 21595 degrees of freedom
Multiple R-squared: 0.7276, Adjusted R-squared: 0.7274
F-statistic: 3393 on 17 and 21595 DF, p-value: < 2.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